It is proved that the solution to the initial value problem ${\partial }_{t}u=\partial {²}_{x}u+u²$, u(0,x) = 1/(1+x²), does not belong to the Gevrey class $G^s$ in time for 0 ≤ s < 1. The proof is based on an estimation of a double sum of products of binomial coefficients.

2000 Mathematics Subject Classification: 26A33, 33C60, 44A15, 35K55Denoting by Dα0|t the time-fractional derivative of order α (α ∈ (0, 1)) in the sense of Caputo, and by ∆H the Laplacian operator on the (2N + 1) - dimensional Heisenberg group H^N, we prove some nonexistence results for solutions to problems of the type Dα0|tu − ∆H(au) >= |u|^p, Dα0|tu − ∆H(au) >= |v|^p, Dδ0|tv − ∆H(bv) >= |u|^q, in H^N × R+ , with a, b ∈ L ∞ (H^N × R+). For α = 1 (and δ = 1 in the case of two inequalities),...

We study existence and approximation of non-negative solutions of partial differential equations of the type$${\partial }_{t}u-div\left(A(\nabla \left(f\left(u\right)\right)+u\nabla V)\right)=0\text{in}(0,+\infty )\times {ℝ}^n,(0.1)$$where $A$ is a symmetric matrix-valued function of the spatial variable satisfying a uniform ellipticity condition, $f:[0,+\infty )\to [0,+\infty )$ is a suitable non decreasing function, $V:{ℝ}^n\to ℝ$ is a convex function. Introducing the energy functional $\phi \left(u\right)={\int }_{{ℝ}^n}F\left(u\left(x\right)\right)\mathrm{d}x+{\int }_{{ℝ}^n}V\left(x\right)u\left(x\right)\mathrm{d}x$, where $F$ is a convex function linked to $f$ by $f\left(u\right)=u{F}^{\text{'}}\left(u\right)-F\left(u\right)$, we show that $u$ is the “gradient flow” of $\phi$ with respect to the 2-Wasserstein distance between probability measures on the space...

We study existence and approximation of non-negative solutions of partial differential equations of the type 
 $${\partial }_{t}u-div\left(A(\nabla \left(f\left(u\right)\right)+u\nabla V)\right)=0\text{in}(0,+\infty )\times {ℝ}^n,(0.1)$$ where A is a symmetric matrix-valued function of the spatial variable satisfying a uniform ellipticity condition, $f:[0,+\infty )\to [0,+\infty )$ is a suitable non decreasing function, $V:{ℝ}^n\to ℝ$ is a convex function. Introducing the energy functional $\phi \left(u\right)={\int }_{{ℝ}^n}F\left(u\left(x\right)\right)\mathrm{d}x+{\int }_{{ℝ}^n}V\left(x\right)u\left(x\right)\mathrm{d}x$, where F is a convex function linked to f by $f\left(u\right)=u{F}^{\text{'}}\left(u\right)-F\left(u\right)$, we show that u is the “gradient flow” of ϕ with respect to the 2-Wasserstein distance between probability measures on the space...

We prove an existence result for a class of parabolic problems whose principal part is the $p$-Laplace operator or a more general Leray-Lions type operator, and featuring an additional first order term which grows like ${\left|\nabla u\right|}^p$. Here the spatial domain can have infinite measure, and the data may be not regular enough to ensure the boundedness of solutions. As a consequence, solutions are obtained in a class of functions with exponential integrability. An existence result of bounded solutions is also given...